Roderick MacPhee - cp's OEIS frontend

User: Roderick MacPhee

Roderick MacPhee has authored 2 sequences.

A289206 Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3.

1, 2, 5, 6, 12, 13, 15, 16, 32, 33, 35, 39, 40, 42, 56, 81, 84, 85, 88, 90, 93, 94, 108, 109, 113, 115, 116, 159, 189, 207, 208, 222, 223, 232, 235, 240, 243, 244, 249, 250, 252, 259, 267, 271, 289, 304, 314, 318, 325, 340, 342, 397, 504, 508, 511, 531, 549

Offset: 1

Roderick MacPhee, Jun 28 2017

nonn

By avoiding arithmetic progressions, at most 2/3 of the numbers up to a(n) are in the sequence. The sequence doesn't contain 3 consecutive powers in arithmetic progression for any base c.

Where a(n)+1 = a(n+1): 1, 3, 5, 7, 9, 12, 17, 21, 23, 26, 30, 32, 37, 39, etc. - Robert G. Wilson v, Jul 02 2017

			5 is in the sequence because 1,2,5 is neither an arithmetic progression nor a geometric progression.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Math StackExchange, A sequence that avoids both arithmetic and geometric progression (2014)

Cf. A000452, A003278, A005836, A224853, A225571.

{my(a=[1,2]);
for(x=3,100,
if(#select(r->#select(q->q==2*r,b)==0,b=vecsort(apply(r->x-r,a)))==#a && #select(r->#select(q->q==r^2,b)==0,b=vecsort(apply(r->x/r,a)))==#a,a=concat(a,x)));a
}

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, my(avoid=List(),t,last=v[k-1]); for(i=2,k-1, for(j=1,i-1, t=2*v[i]-v[j]; if(t>last, listput(avoid, t)); if(denominator(t=v[i]^2/v[j])==1 && t>last, listput(avoid,t)))); avoid=Set(avoid); for(i=v[k-1]+1,v[k-1]+#avoid+1, if(!setsearch(avoid,i), v[k]=i; break))); v \\ Charles R Greathouse IV, Jun 29 2017

a(n) >= 3n/2 for n > 2.

More terms from Alois P. Heinz, Jun 28 2017

A173912 Numbers x that when put through Lucas-Lehmer tests give a residue that has a digital root of 0 or 9.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 31, 33, 51, 61, 71, 89, 101, 107, 127, 139, 191, 271, 273, 305, 331, 347, 351, 367, 397, 405, 407, 427, 435, 457, 467, 489, 521, 525, 539, 543, 549, 559, 565, 577, 583, 589, 597, 601, 607, 611, 613, 617, 619, 641, 643, 661, 693, 717, 729, 787, 793, 809, 817, 819, 837, 871, 879, 891, 899, 983, 987, 991

Offset: 1

Roderick MacPhee, Nov 26 2010

The PARI code uses a function that assumes 0 has a digital root of 9.

Note: since I allowed 0 to count as having digital root 9, all Mersenne prime exponents > 2 will be a subsequence of this sequence.

lucaslehmer2Q[p_] := Module[{s = 4, x}, For[x = 1, x <= p-2, x++, s = Mod[s^2 - 2, 2^p - 1]; If[x == p-2 && sumdigits1[s] == 9, Return[True]]]; False];
sumdigits1[n_] := If[Mod[n, 9] != 0, Mod[n, 9], 9];
Select[Range[1000], lucaslehmer2Q] (* Jean-François Alcover, Sep 28 2020, after PARI *)

lucaslehmer2(p) = s=4; for(x=1, p-2, s=(s^2-2)%(2^p-1)); if(x=p-2 && sumdigits1(s)==9, print1(p", "))
sumdigits1(n)=if(n%9!=0,n%9,9)
for(x=1,1000,lucaslehmer2(x))

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User: Roderick MacPhee

A289206 Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3.

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A173912 Numbers x that when put through Lucas-Lehmer tests give a residue that has a digital root of 0 or 9.

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